OFFSET
1,2
COMMENTS
This is called the "Simple Zeckendorf diagonal sequence in two dimensions". - N. J. A. Sloane, Dec 11 2020
The array T(i,j) is constructed as follows, by filling in the array by downward antidiagonals, in this order:
1, 2, 4, 7, ...
3, 5, 8, ...
6, 9, ...
10, ...
Start with T(1,1) = 1.
Look at the next empty square (on the succession of antidiagonal paths). Iterate through the natural numbers, starting at the last number that we accepted. When we consider a number k, if we can get k as a sum of existing numbers that are on a path in the array starting at any filled square and always moves up and to the left, skip k, and consider k+1. Otherwise, enter k in the square.
For example, T(3,1) is not 6 because we can get 6 as the sum of 5 and 1. So T(3,1) = 7.
T(3,2) is not 10, because we can get 10 as 9+1, and it not 11 because of 9+2. So T(3,2) = 12.
REFERENCES
Borade, N., Cai, D., Chang, D. Z., Fang, B., Liang, A., Miller, S. J., & Xu, W. (2019). Gaps of Summands of the Zeckendorf Lattice. Fib. Q., 58:2 (2020), 143-156. See the array y_{i,j}, although in the present entry the order of the rows has been reversed.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..630 (first 35 rows flattened)
Neelima Borade, Dexter Cai, David Z. Chang, Bruce Fang, Alex Liang, Steven J. Miller, and Wanqiao Xu, Gaps of Summands of the Zeckendorf Lattice, arXiv:1909.01935 [math.NT], 2019.
Chen, E., Chen, R., Guo, L., Jiang, C., Miller, S. J., Siktar, J. M., & Yu, P. (2018). Gaussian Behavior in Zeckendorf Decompositions From Lattices, arXiv preprint arXiv:1809.05829. Also Fib. Q., 57:5 (2019), 201-212.
Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice, arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.
Rémy Sigrist, PARI program for A335154
EXAMPLE
The array begins:
1, 2, 4, 8, 16, 29, 54, 90, 159, ...
3, 5, 9, 17, 30, 56, 93, 160, ...
7, 12, 20, 33, 59, 100, 171, ...
14, 24, 40, 66, 107, 184, ...
28, 48, 74, 123, 198, ...
50, 82, 139, 230, ...
84, 155, 259, ...
157, 263, ...
280, ...
...
The first few antidiagonals are:
1;
2, 3;
4, 5, 7;
8, 9, 12, 14;
16, 17, 20, 24, 28;
29, 30, 33, 40, 48, 50;
...
In Chen et al. (2019) the array appears as follows:
...
280, ...
157, 263, ...
84, 155, 259, ...
50, 82, 139, 230, ...
28, 48, 74, 123, 198, ...
14, 24, 40, 66, 107, 184, ...
7, 12, 20, 33, 59, 100, 171, ...
3, 5, 9, 17, 30, 56, 93, 160, ...
1, 2, 4, 8, 16, 29, 54, 90, 154, ...
PROG
(PARI) \\ See Links section.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 02 2020
EXTENSIONS
More terms from Rémy Sigrist, Jun 10 2020
STATUS
approved